Advances in genomics have led to calls for developing population-based preventive genomic sequencing (PGS) programs with the goal of identifying genetic health risks in adults without known risk factors. One critical issue for minimizing the harms and maximizing the benefits of PGS is determining the kind and degree of control individuals should have over the generation, use, and handling of their genomic information. In this article we examine whether PGS programs should offer individuals the opportunity to selectively opt-out of the sequencing or analysis of specific genomic conditions (the menu approach) or whether PGS should be implemented using an all-or-nothing panel approach. We conclude that any responsible scale up of PGS will require a menu approach that may seem impractical to some, but which draws its justification from a rich mix of normative, legal, and practical considerations.